Yes, it’s possible. Get the SeaJet ferry from Athens to Mykonos. It should arrive at 9:30am or shortly after. You’ll have to hurry to store you’re luggage at the Sea and Sky travel agency across from the Old Port and then buy tickets for Delos. The last ferry to Delos leaves at 11am or maybe 11:30am. Ferry to Delos is about 30 minutes. Last ferry back is at 3pm which will give you plenty of time before the last ferry of the day to Naxos (at 6:15pm on SeaJet). All the ferries mention here use the Old Port. Don’t book any ferries that use the New Port or you’ll have trouble making the connections.
We are having trouble deciding on another island to go to besides Santorini (we both want to go there). I was hoping you might be able to make a suggestion. We are not really into late night partying/night life. We LOVE good food..quite possibly the most important item on our list. We also like to hike, my husband is very into history, we love beer/wine, we could definitely be into in a less populated/touristy type spot. Gorgeous beaches and great views are also a plus.